Numerische Mathematik

, Volume 143, Issue 4, pp 781–795 | Cite as

Generalized Gaffney inequality and discrete compactness for discrete differential forms

  • Juncai He
  • Kaibo HuEmail author
  • Jinchao Xu


We prove generalized Gaffney inequalities and the discrete compactness for finite element differential forms on s-regular domains, including general Lipschitz domains. In computational electromagnetism, special cases of these results have been established for edge elements with weakly imposed divergence-free conditions and used in the analysis of nonlinear and eigenvalue problems. In this paper, we generalize these results to discrete differential forms, not necessarily with strongly or weakly imposed constraints. The analysis relies on a new Hodge mapping and its approximation property. As an application, we show \(L^{p}\) estimates for several finite element approximations of the scalar and vector Laplacian problems.

Mathematics Subject Classification

65N30 65N12 58J10 35D30 



The authors are grateful to Prof. Ralf Hiptmair for several helpful suggestions.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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